A simple aperiodic random line moiré can be obtained by
superposing two identical layers comprising randomly spaced horizontal parallel
lines, where one of the layer patterns is slightly scaled down vertically.

The layer which is scaled down will be denoted in our
example as the revealing layer and the layer which is not scaled as the base
layer.

Superposition of such two layers outlines a light horizontal
moiré line. In contrast to periodic patterns, there will be only single
appearance of the moiré line. By moving the revealing layer up vertically, the
moiré line will move up at a higher velocity.

Let be the scaling factor, be the speed of the moiré line, and be the speed of the revealing layer. The
optical speedup factor is equal to:

We assume that the base layer is immobile, i.e. .

If all parallel lines of the base layer are inclined by degrees, and as before the revealing
layer pattern is the scaled down copy of the base layer pattern, where the
scaling is vertical and the scaling factor is denoted as , then the following equation will hold:

Superposition of two such patterns will form a moiré line,
where , since:

Therefore the moiré light line will appear again horizontally.
The optical speedup factor will not change.

The animation below shows an example of random line moiré.
The base layer lines are inclined by 20 degrees. The scaling factor k is
equal to 11/12. The revealing layer is moving up vertically at a slow speed.
The white moiré line of the superposition image moves up 12 times faster
(according to the equation for the optical speedup).